Is it possible to snap 1 or more vertices on a face?
You can do this with the snap tool.

First, modify snapping behavior to Snap to Face like so :

 (toolbar on the top of the viewport screen)



In edit mode, select all the vertices you want to snap to the faces.

Then go into orthographic mode (numpad 5), align view to the "x" axis (numpad 3)



Type G (to move the vertices) and X to constrain to the x axis, and move your mouse to the right a fair bit, Hold CTRL (to force snapping), then click. You will not see any change in the viewport but if you orbit around your objects you will have this result :